Editing Polyhedron (section)

=== By elements of polyhedron ===
All the elements (vertex, face, and edge) that can be superimposed on each other by symmetries are said to form a [[Symmetry orbit#Orbits and stabilizers|symmetry orbit]]. If these elements lie in the same orbit, the figure may be transitive on the orbit. Individually, they are [[isohedral]] (or face-transitive, meaning the symmetry transformations involve the polyhedra's faces in orbit),<ref name=mclean>{{citation
 | last = McLean | first = K. Robin
 | year = 1990
 | title = Dungeons, dragons, and dice
 | journal = [[The Mathematical Gazette]]
 | volume = 74 | issue = 469 | pages = 243–256
 | doi = 10.2307/3619822
 | jstor = 3619822
 | s2cid = 195047512
}} See p. 247.</ref>{{efn|1=The topological property of an isohedral polyhedra can be represented by a [[face configuration]]. All five [[Platonic solids]] and thirteen [[Catalan solid]]s are isohedra, as well as the infinite families of [[trapezohedra]] and [[bipyramid]]s. Some definitions of isohedra allow geometric variations including concave and self-intersecting forms.}} [[isotoxal]] (or edge-transitive, which involves the edge's polyhedra),<ref name=grunbaum-1997>{{citation
 | last = Grünbaum | first = Branko | authorlink = Branko Grünbaum 
 | year = 1997
 | title = Isogonal Prismatoids
 | journal = Discrete & Computational Geometry
 | volume = 18 | issue = 1 | pages = 13&ndash;52
 | doi = 10.1007/PL00009307
}}</ref> and [[isogonal figure|isogonal]] (or vertex-transitive, which involves the polyhedra's vertices). For example, the [[cube]] in which all the faces are in one orbit and involving the rotation and reflections in the orbit remains unchanged in its appearance; hence, the cube is face-transitive. The cube also has the other two such symmetries.<ref name=senechal>{{citation
 | last = Senechal | first = Marjorie
 | year = 1989
 | contribution = A Brief Introduction to Tilings
 | contribution-url = https://books.google.com/books?id=OToVjZW9CKMC&pg=PA12
 | editor-last = Jarić | editor-first = Marko
 | title = Introduction to the Mathematics of Quasicrystals
 | publisher = [[Academic Press]]
 | page = 12
}}</ref>

[[File:Hexahedron.svg|thumb|upright=0.6|The [[cube]] is a [[regular polyhedron]], because its faces, edges, and vertices are transitive to another, and the appearance is unchanged.]]
When three such symmetries belong to a polyhedron, it is known as a [[regular polyhedron]].<ref name=senechal /> There are nine regular polyhedra: five [[Platonic solid]]s (cube, [[regular octahedron|octahedron]], [[regular icosahedron|icosahedron]], [[regular tetrahedron|tetrahedron]], and [[regular dodecahedron|dodecahedron]]&mdash;all of which have regular polygonal faces) and four [[Kepler&ndash;Poinsot polyhedron]]s. Nevertheless, some polyhedrons may not possess one or two of those symmetries:
* A polyhedron with vertex-transitive and edge-transitive is said to be a [[quasiregular polyhedron|quasiregular]], although they have regular faces, and its dual is face-transitive and edge-transitive.
* A vertex- but not edge-transitive polyhedron with regular polygonal faces is said to be a [[Semiregular polyhedron|semiregular]].{{efn|1=This is one of several definitions of the term, depending on the author. Some definitions overlap with the quasi-regular class.}} and such polyhedrons are [[prism (geometry)|prisms]] and [[antiprism]]s. Its dual is face-transitive but not vertex-transitive, and every vertex is regular.
* A polyhedron with regular polygonal faces and vertex-transitive is said to be [[Uniform polyhedron|uniform]]. This class may be subdivided into a regular, quasi-regular, or semi-regular polyhedron, and may be convex or starry. The dual is face-transitive and has regular vertices but is not necessarily vertex-transitive. The uniform polyhedra and their duals are traditionally classified according to their degree of symmetry, and whether they are [[Convex polyhedron|convex]] or not.
* A face- and vertex-transitive (but not necessarily edge-transitive) polyhedra is said to be [[Noble polyhedron|noble]]. The regular polyhedra are also noble; they are the only noble uniform polyhedra. The duals of noble polyhedra are themselves noble.

Some polyhedra may have no [[reflection symmetry]] such that they have two enantiomorph forms, which are reflections of each other. Such symmetry is known for having [[Chirality (mathematics)|chirality]]. Examples include the [[snub cuboctahedron]] and [[snub icosidodecahedron]].